\lim_{x\to\infty}x^2\int^{x}_{0}e^{t^3-x^3}dt equals

KVPY SX SB Exam 2016 Question Paper

Question : 12 of 120

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\lim↙{x→∞}x^2∫↖{x}↙{0}e^{t^3-x^3}dt

equals

Solution:

\lim↙{x→∞}{{x^2∫↖{x}↙{0}e^{t^3 }}dt}/e^{x^3}

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⇒\lim↙{x→∞}{2x∫↖{x}↙{0}e^{t^3}+x^2e^{x^3}}/{3x^2e^{x^3}}

⇒\lim↙{x→∞}{2 ∫↖{x}↙{0}e^{t^3}+x e^{x^3}}/{3x e^{x^3}}

⇒\lim↙{x→∞} {2e^{x^3}+e^{x^3}+3x^3e^{x^3}}/{3e^{x^3}+9x^3e^{x^3}}

⇒\lim↙{x→∞}{3+3x^3}/{3+9x^3}=1/3

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